Download Rigid Body Dynamics (unconstrained)

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Transcript 
Rigid Body Dynamics
(unconstrained)
Simulation Basics
State vector of a single particle
Change of Y(t) over time
Solved by any ODE solver (Euler, Runge-Kutta, etc.)
Rigid Body Concepts
• Body space
– Origin: center of mass
• p0: an arbitrary point on the
rigid body, in body space.
– Its world space location p(t)
• Spatial variables of the rigid
body: 3-by-3 rotation matrix
R(t) and x(t)
The Rotation Matrix
• Three columns of R(t)
correspond to the axes
of the body-space in
the world space
Linear and Angular Velocity
• How are R(t) and w(t)
related?
R(t) and w(t)
R(t) and w(t)
Velocity of a Particle
Force and Torque
Single particle
Linear Momemtum
Center of Mass
Angular Momemtum
Inertia Tensor
Inertia Tensor
Equation of Motion
Inertia Tensor of a Block
Inertia Tensor Table (ref)
Uniform Force Field
No effect on the angular momentum
The Football in Flight (ref)
Gravity does not exert torque
Angular momentum stays the same
Using Quaternion
quaternion
multiplication
Unit quaternion
as rotation
quaternion
derivative
Equation of
motion
Computing Qdot
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